Action Research Project
Data Collection Methods:
Data were collected using the GMF survey (appended to this section), which focused on the following:
Items 1-7: Recall and evaluation of the process of annotation as learned in Mr. Roth’s 05-06 Inquiry Skills class.
Items 8-9, 12-17, 19-20: Recall and evaluation of the process of using the GMF.
Items 10-11, 18: Relating annotation and the GMF.
After collection, the average response for each survey item was calculated for each class, as well as for the entire set of participants. Furthermore, any additional comments made by participants were transcribed to a single document (appended).
Download: GMF Survey
Title:
“Structured Metacognition and the Annotative Process”
Statement of Problem:
Purpose
The purpose of this study is to examine the role of structured metacognitive practice for students of high school mathematics at Jim Hill High School. For the purposes of this study, “structured metacognitive practice” will be defined as/represented by the use of annotative explanation as a means of supplementing student work.
Background
About once a week – with surprising regularity – a student in one of my math classes at Jim Hill High School in Jackson, MS will inevitably raise his/her hand and proclaim, “Mr. Molina, I’m confused” or “Mr. MoMo” – which is my unavoidable nickname – “I don’t get it.” To this I will respond, “Great!” to which the student will counter-respond, “why is that great?” to which I will counter-counter-respond, “it’s not great that you’re confused; it’s great that you want to talk about it.” Needless to say, I put a high premium on reflection and self-awareness in the learning process.
As a teacher of mathematics – a discipline that holds its heritage as the Queen of Sciences in one hand and a paranoia about ever-approaching irrelevance in the other – I’m concerned more about my students being thinkers than I am about them being learners. That is, I honestly couldn’t care less if my students have been exposed to every nook and cranny of the mathematical content deemed appropriate to a high school student – if they haven’t first and foremost developed a working relationship with the way that they best interact with any system of knowledge. Furthermore, the relative abstract purity of mathematical discourse makes it an excellent environment for a student to reflect on how he/she engages with and adapts to new systems of information. So, I’ve made continuous efforts as an instructor to integrate a process of academic self-reflection – thus metacognition – into my classroom.
In my mind, one of the more successful metacognitive routines that I’ve developed has been what I call the “General Metacognitive Form” (GMF) – a uselessly wordy title for a sheet that my students more appropriately understand to be “the corrections sheet that we have to do after tests.” Its creation – in draft form – was relatively straightforward: I wanted to put together a worksheet that abstracted the process of verbalizing the roots of a mistake, in the hopes that it would help patch up the thinking process and prevent further mistakes. Furthermore, I wanted the abstract nature of the sheet to be highly transferable – so that a student could use it in any appropriate mathematical context (test, quiz, homework, discussion, project, etc.).
Though this seemed a humble beginning for an unassuming worksheet that was nevertheless necessary, I realized at the beginning of the 2006-2007 school year (my second year of teaching at Jim Hill) that my Algebra II students – 10th graders in Jim Hill’s IB Programme (IB) – had been trained by a colleague and friend of mine – Jacob Roth – to perform a task that was incredibly ripe for transferal into a more informed engagement with the GMF than I had originally envisioned. This task was the process of annotation.
Jake had raved about the importance of annotation all throughout the previous school year – when these 10th grade Math students of mine were then 9th grade English students of his. He would photocopy endless poems and short stories and demand that the remaining white space on the page be filled with definitions, questions, ideas, etc. Jake thought it was important that students learn to read with a pen – through this process they could engage in a text in a way that would make it their own. Jake said he knew annotation was sinking in when he would say “this sentence is important,” or “this word means” and – in a flash – pens and highlighters would be out to nail down whatever was necessary.
In the fall of the 06-07 school year, while my Algebra II class was discussing the Edna St. Vincent Milay poem, “Euclid Alone Has Looked on Beauty Bare,” as well as excerpts form Alfred North Whitehead’s Introduction to Mathematics and Martin Gardner’s Mathematical Puzzles and Diversions, I noticed that Jake’s annotation instruction had really stuck with his kids – now mine. Sure enough, when I said “this is important” or “this word means,” pens flew out. I realized at this point that the task of annotating – hardwired into my students after a year with Jake – could enrich the use of the GMF to correct student work. Very simply, the idea of “correcting” could be replaced by “annotating.” This linguistic turn is especially pleasing for a teacher who gets excited when a student is ready to proclaim that he/she is confused; “correcting” has the pejoratively reflective sense of a fault or error, while “annotating” has a more productive and understanding intent; “annotating” is truly self-reflective, while “correcting” maintains an air of self-criticism.
In the past semester, filling out the GMF became a process of literally identifying (by circling) an error, explaining (by annotating) what was going on within the text of the problem, and then trying the problem again. This activity was intended to be a re-reading of a thought process that has been put on paper, and – in my mind – served to validate the fact that this was a thought process, not just a random output of numbers. In another teacherly outburst that never fails to confuse my students, I often champion the footwork behind a reasonable mistake, stating that, “the only thing that’s wrong with it is that it’s wrong.” The GMF gives a student the opportunity to own this reasonable process and improve it at the same time through annotation, instead of disowning and devaluing their own work through the process of correction.
This study is intended to look at how my students view the process of annotation, how they view the GMF, and how they view the connection between. The desire is always that a structured act of metacognition – in this case, the process of annotation turned upon one’s own work – is beneficial to the development of students as independent and adaptive thinkers.
Analysis of Findings:
Results
The preceding graph shows the average student response for each survey item. Each item used the following scale: 1=Strongly Disagree, 3=Indifferent, 5=Strongly Agree.
From the graph, it is clear to note a distinct pattern in student responses. Some observations:
Highest cumulative average: Item 1, “I am familiar with the process of annotation.” (4.9)
Lowest cumulative average: Item 7, “I frequently annotate nonacademic texts.” (2.2)
Cumulative averages at or above 4.0:
Item 1, “I am familiar with the process of annotation.” (4.9)
Item 2, “I can remember learning to use the process of annotation in Mr. Roth’s class.”(4.8)
Item 13, “Understanding my mistakes is important.” (4.8)
Item 10, “In the GMF I am asked to annotate my own work.”(4.6)
Item 9, “I remember learning to use the GMF in Mr. Molina’s class.” (4.5)
Item 8, “I am familiar with the GMF.”(4.4)
Item 11, “I use annotation skills similar to the ones that I developed in Mr. Roth’s class to fill out the GMF.” (4.3)
Item 12, “I find the GMF beneficial to the process of understanding my mistakes.” (4.1)
Item 4, “I found annotation beneficial to the process of understanding what I was reading.” (4)
Cumulative averages at or below 3.0:
Item 6, “I frequently annotate texts in other classes even when not prompted to do so.” (3.0)
Item 5, “I frequently annotate texts in my English class even when not prompted to do so.” (2.8)
Item 16, “I annotate my math work even when not prompted to do so.” (2.6)
Item 7, “I frequently annotate nonacademic texts.” (2.2)
Discussion
1. Relating the survey results to the areas of focus
Items 1-7: Recall and evaluation of the process annotation as learned in Mr. Roth’s 05-06 Inquiry Skills class.
The data in items 1-7 provide perhaps the clearest picture of the whole survey. It is obvious that Mr. Roth’s students know what annotation is (Item 1, 4.9), remember learning it in Jake’s class (Item 2, 4.8) and found it useful to understanding a text (Item 3, 3.9). Furthermore, a clear and reasonable distancing of themselves from the annotation process occurs when students are asked if they annotate texts at different transferable distances from their original context (Item 5, 2.8, Item 6, 3, and Item 7, 2.2) –which seems to illustrate some level of honesty involved in their evaluation.
Items 8-9, 12-17, 19-20: Recall and evaluation of the process of using the GMF
With the GMF, it is clear once again that the students taking the survey both know what it is (Item 8) and remember learning how to use it in my class (Item 9). Also, my students clearly seem to value both the process of learning from their mistakes (Item 13, 4.8), and find the GMF beneficial to that process (Item 12, 4.1). However, the effectiveness of the GMF in preventing new mistakes (Item 14, 3.8, and Item 15, 3.5) appears less convincing, though hopeful – as students seem to recommend that the form continue to be used (Item 19, 3.9). Lastly, the transfer of the skills used in the GMF into new contexts falls off nearly as much (Item 16 2.6) as the process of annotation.
Items 10-11, 18: Relating annotation and the GMF.
Students appear to see a strong connection between the GMF and the process of annotation (Item 10, 4.6), specifically the annotation skills developed in Mr. Roth’s class (Item 11, 4.3), even though the word “annotation” is never used in the GMF, and the two have only been connected by my explanation about how the GMF functions (i.e. “we will use this form to annotate our own work…”). Furthermore, students seem nearly indifferent about whether or not the GMF would be as effective had they not learned how to annotate in Mr. Roth’s class (Item 18, 3.3), though this may indicate either the strong connection between annotation and the GMF, or confusion about the Item’s wording.
Additional Comments
The “additional comments” section of the GMF survey elicited some interesting responses, ranging (not surpisingly) from the negative:
“I think Mr. Molina and Mr. Roth need to learn what subjects their [sic] teaching and quit if they don’t like it.”
To the bizarre:
“Personally… I could really care less! I sleep in class, so I miss half of what’s going on anyway! I would prefer that we not do this anymore because it’s a waste of my time and it shows how simple and meaningless of a life you live! Love ya [sic] much! ~Duece, Duece~! Long live tha [sic] pimp! Phrrr! Bird call! M.O.B. to tha [sic] grave. I’m out my nigga… Weezy F. MoMo! Please say the MoMo!”
To the flattering:
“I think that Mr. MoMo is quite a good teacher. He is always upbeat and that helps me and others to pay attention in class. He is great with explaining different things and making it [sic] easier for me to comprehend.”
To the constructively critical:
“I think it would be best to go over the concepts to getting [sic] an answer and spend more time learning how to do things and then we won’t have to use the GMF.”
To the profound:
“I think I’ve often annotated before Mr. Roth’s class, but mostly I think it was in my head. I think learning to annotate and actually continuing the process on into a new class – instead of starting a new helpful process and stopping it (which is often done) – is very helpful, and I enjoy it.”
2. Limitations and shortcomings
It must be admitted that the GMF survey itself was hastily put together, even though it was able to produce analysis-worthy data. Changes could easily be made in the wording of many of the items – to prevent confusion about what exactly is being asked. Looking over the survey, Mr. Roth noted that the language of Item 4 (the word “analyze”), Item 12 (the word “beneficial”), Item 15 (general wordiness), and Item 18 (general wordiness) could be improved. Furthermore, the number of items testing the areas of focus could be more equally distributed, and – perhaps most importantly – each item could have a positive and negative counterpart, to correct for possible bias.
Implications for Instruction/Education:
This study reinforced what was obvious: (a) Jake’s students are familiar with annotation and find it beneficial to understanding what they’re reading, (b) my students are familiar with the GMF and find it beneficial to understanding their mistakes, and (c) these students can see a connection between their annotation skills and the GMF activity. Strangely, the negative trends in student responses seem to provide validity to the positive ones listed above (i.e. the survey-takers weren’t merely answering in the way that they thought I wanted them to). Furthermore, it is exactly these negative trends that seem to be the most seductive for a teacher interested in helping students become better thinkers. In this light, the question now becomes: how do we get students to see the value of independently transferring those analytical and metacognitive practices that they find beneficial in our classrooms? While it is not surprising that students seemed to annotate less and less as texts moved from current English classes, to other classes, to nonacademic texts – and it is similarly not surprising that students don’t seem to be annotating their own math work when the GMF is not around – this is exactly what we want them to do once they readily acknowledge that annotation and things like the GMF are in fact valuable. Perhaps it will take some meta-metacognition, some thinking with our students about how they think about thinking, to help them make that next leap: from receptive learners to productive thinkers.
